Best Known (146, 157, s)-Nets in Base 4
(146, 157, 7235174)-Net over F4 — Constructive and digital
Digital (146, 157, 7235174)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (28, 33, 524294)-net over F4, using
- net defined by OOA [i] based on linear OOA(433, 524294, F4, 5, 5) (dual of [(524294, 5), 2621437, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(433, 1048589, F4, 5) (dual of [1048589, 1048556, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(432, 1048588, F4, 5) (dual of [1048588, 1048556, 6]-code), using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- linear OA(431, 1048576, F4, 5) (dual of [1048576, 1048545, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(421, 1048576, F4, 3) (dual of [1048576, 1048555, 4]-code or 1048576-cap in PG(20,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(411, 12, F4, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,4)), using
- dual of repetition code with length 12 [i]
- linear OA(41, 12, F4, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(4) ⊂ Ce(2) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(432, 1048588, F4, 5) (dual of [1048588, 1048556, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(433, 1048589, F4, 5) (dual of [1048589, 1048556, 6]-code), using
- net defined by OOA [i] based on linear OOA(433, 524294, F4, 5, 5) (dual of [(524294, 5), 2621437, 6]-NRT-code), using
- digital (113, 124, 6710880)-net over F4, using
- trace code for nets [i] based on digital (20, 31, 1677720)-net over F256, using
- net defined by OOA [i] based on linear OOA(25631, 1677720, F256, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25631, 8388601, F256, 11) (dual of [8388601, 8388570, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25631, 8388601, F256, 11) (dual of [8388601, 8388570, 12]-code), using
- net defined by OOA [i] based on linear OOA(25631, 1677720, F256, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
- trace code for nets [i] based on digital (20, 31, 1677720)-net over F256, using
- digital (28, 33, 524294)-net over F4, using
(146, 157, large)-Net over F4 — Digital
Digital (146, 157, large)-net over F4, using
- 42 times duplication [i] based on digital (144, 155, large)-net over F4, using
- t-expansion [i] based on digital (139, 155, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4155, large, F4, 16) (dual of [large, large−155, 17]-code), using
- 11 times code embedding in larger space [i] based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 11 times code embedding in larger space [i] based on linear OA(4144, large, F4, 16) (dual of [large, large−144, 17]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4155, large, F4, 16) (dual of [large, large−155, 17]-code), using
- t-expansion [i] based on digital (139, 155, large)-net over F4, using
(146, 157, large)-Net in Base 4 — Upper bound on s
There is no (146, 157, large)-net in base 4, because
- 9 times m-reduction [i] would yield (146, 148, large)-net in base 4, but